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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials


Authors: Alexander I. Aptekarev, Sergey A. Denisov and Maxim L. Yattselev
Journal: Trans. Amer. Math. Soc. 373 (2020), 875-917
MSC (2010): Primary 42C05, 47B36
DOI: https://doi.org/10.1090/tran/7959
Published electronically: October 28, 2019
MathSciNet review: 4068253
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Abstract: We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb {Z}_+$ to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.


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Additional Information

Alexander I. Aptekarev
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Science, Moscow, Russian Federation
MR Author ID: 192572
Email: aptekaa@keldysh.ru

Sergey A. Denisov
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
MR Author ID: 627554
Email: denissov@math.wisc.edu

Maxim L. Yattselev
Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202
MR Author ID: 789878
Email: maxyatts@iupui.edu

Received by editor(s): June 27, 2018
Received by editor(s) in revised form: January 15, 2019
Published electronically: October 28, 2019
Additional Notes: The research of the first author was carried out with support from a grant of the Russian Science Foundation (project RScF-14-21-00025)
The work of the second author done in the last section of the paper was supported by a grant of the Russian Science Foundation (project RScF-14-21-00025), and his research on the rest of the paper was supported by the grant NSF-DMS-1464479 and a Van Vleck Professorship Research Award
The research of the third author was supported in part by a grant from the Simons Foundation, CGM-354538
Article copyright: © Copyright 2019 American Mathematical Society